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Theorem tanatan 23831
Description: The arctangent function is an inverse to  tan. (Contributed by Mario Carneiro, 2-Apr-2015.)
Assertion
Ref Expression
tanatan  |-  ( A  e.  dom arctan  ->  ( tan `  (arctan `  A )
)  =  A )

Proof of Theorem tanatan
StepHypRef Expression
1 atancl 23793 . . 3  |-  ( A  e.  dom arctan  ->  (arctan `  A )  e.  CC )
2 2efiatan 23830 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  1 ) )
32oveq1d 6316 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  +  1 )  =  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  +  1 ) )
4 2mulicn 10836 . . . . . . . 8  |-  ( 2  x.  _i )  e.  CC
54a1i 11 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 2  x.  _i )  e.  CC )
6 atandm 23788 . . . . . . . . 9  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/= 
_i ) )
76simp1bi 1020 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  A  e.  CC )
8 ax-icn 9598 . . . . . . . 8  |-  _i  e.  CC
9 addcl 9621 . . . . . . . 8  |-  ( ( A  e.  CC  /\  _i  e.  CC )  -> 
( A  +  _i )  e.  CC )
107, 8, 9sylancl 666 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( A  +  _i )  e.  CC )
11 subneg 9923 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  _i  e.  CC )  -> 
( A  -  -u _i )  =  ( A  +  _i ) )
127, 8, 11sylancl 666 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( A  -  -u _i )  =  ( A  +  _i ) )
136simp2bi 1021 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  A  =/=  -u _i )
148negcli 9942 . . . . . . . . . 10  |-  -u _i  e.  CC
15 subeq0 9900 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  -u _i  e.  CC )  ->  ( ( A  -  -u _i )  =  0  <->  A  =  -u _i ) )
1615necon3bid 2682 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  -u _i  e.  CC )  ->  ( ( A  -  -u _i )  =/=  0  <->  A  =/=  -u _i ) )
177, 14, 16sylancl 666 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( ( A  -  -u _i )  =/=  0  <->  A  =/=  -u _i ) )
1813, 17mpbird 235 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( A  -  -u _i )  =/=  0 )
1912, 18eqnetrrd 2718 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( A  +  _i )  =/=  0 )
205, 10, 19divcld 10383 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  /  ( A  +  _i ) )  e.  CC )
21 ax-1cn 9597 . . . . . 6  |-  1  e.  CC
22 npcan 9884 . . . . . 6  |-  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  1 )  +  1 )  =  ( ( 2  x.  _i )  / 
( A  +  _i ) ) )
2320, 21, 22sylancl 666 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  +  1 )  =  ( ( 2  x.  _i )  /  ( A  +  _i )
) )
243, 23eqtrd 2463 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  +  1 )  =  ( ( 2  x.  _i )  /  ( A  +  _i ) ) )
25 2muline0 10837 . . . . . 6  |-  ( 2  x.  _i )  =/=  0
2625a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  ( 2  x.  _i )  =/=  0 )
275, 10, 26, 19divne0d 10399 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  /  ( A  +  _i ) )  =/=  0
)
2824, 27eqnetrd 2717 . . 3  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  +  1 )  =/=  0 )
29 tanval3 14175 . . 3  |-  ( ( (arctan `  A )  e.  CC  /\  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  +  1 )  =/=  0 )  ->  ( tan `  (arctan `  A ) )  =  ( ( ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  -  1 )  / 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  +  1 ) ) ) )
301, 28, 29syl2anc 665 . 2  |-  ( A  e.  dom arctan  ->  ( tan `  (arctan `  A )
)  =  ( ( ( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  -  1 )  / 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  +  1 ) ) ) )
312oveq1d 6316 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  -  1 )  =  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  -  1 ) )
3221a1i 11 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  1  e.  CC )
3320, 32, 32subsub4d 10017 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  -  1 )  =  ( ( ( 2  x.  _i )  / 
( A  +  _i ) )  -  (
1  +  1 ) ) )
34 df-2 10668 . . . . . . . 8  |-  2  =  ( 1  +  1 )
3534oveq2i 6312 . . . . . . 7  |-  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  2 )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  (
1  +  1 ) )
3633, 35syl6eqr 2481 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  -  1 )  =  ( ( ( 2  x.  _i )  / 
( A  +  _i ) )  -  2 ) )
3731, 36eqtrd 2463 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  -  1 )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  2 ) )
38 2cn 10680 . . . . . . . 8  |-  2  e.  CC
39 mulcl 9623 . . . . . . . 8  |-  ( ( 2  e.  CC  /\  ( A  +  _i )  e.  CC )  ->  ( 2  x.  ( A  +  _i )
)  e.  CC )
4038, 10, 39sylancr 667 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( A  +  _i ) )  e.  CC )
415, 40, 10, 19divsubdird 10422 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  -  ( 2  x.  ( A  +  _i ) ) )  / 
( A  +  _i ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  ( ( 2  x.  ( A  +  _i ) )  /  ( A  +  _i ) ) ) )
42 mulneg12 10057 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  A  e.  CC )  ->  ( -u 2  x.  A )  =  ( 2  x.  -u A
) )
4338, 7, 42sylancr 667 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( -u
2  x.  A )  =  ( 2  x.  -u A ) )
44 negsub 9922 . . . . . . . . . . . 12  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  +  -u A )  =  ( _i  -  A ) )
458, 7, 44sylancr 667 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  ( _i  +  -u A )  =  ( _i  -  A
) )
4645oveq1d 6316 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( ( _i  +  -u A
)  -  _i )  =  ( ( _i 
-  A )  -  _i ) )
477negcld 9973 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  -u A  e.  CC )
48 pncan2 9882 . . . . . . . . . . 11  |-  ( ( _i  e.  CC  /\  -u A  e.  CC )  ->  ( ( _i  +  -u A )  -  _i )  =  -u A
)
498, 47, 48sylancr 667 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( ( _i  +  -u A
)  -  _i )  =  -u A )
508a1i 11 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  _i  e.  CC )
5150, 7, 50subsub4d 10017 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( ( _i  -  A )  -  _i )  =  ( _i  -  ( A  +  _i )
) )
5246, 49, 513eqtr3rd 2472 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( _i 
-  ( A  +  _i ) )  =  -u A )
5352oveq2d 6317 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( _i  -  ( A  +  _i ) ) )  =  ( 2  x.  -u A
) )
5438a1i 11 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  2  e.  CC )
5554, 50, 10subdid 10074 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( _i  -  ( A  +  _i ) ) )  =  ( ( 2  x.  _i )  -  (
2  x.  ( A  +  _i ) ) ) )
5643, 53, 553eqtr2rd 2470 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  -  ( 2  x.  ( A  +  _i ) ) )  =  ( -u 2  x.  A ) )
5756oveq1d 6316 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  -  ( 2  x.  ( A  +  _i ) ) )  / 
( A  +  _i ) )  =  ( ( -u 2  x.  A )  /  ( A  +  _i )
) )
5854, 10, 19divcan4d 10389 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  ( A  +  _i ) )  /  ( A  +  _i ) )  =  2 )
5958oveq2d 6317 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  (
( 2  x.  ( A  +  _i )
)  /  ( A  +  _i ) ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  2 ) )
6041, 57, 593eqtr3d 2471 . . . . 5  |-  ( A  e.  dom arctan  ->  ( (
-u 2  x.  A
)  /  ( A  +  _i ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  2 ) )
6137, 60eqtr4d 2466 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  -  1 )  =  ( (
-u 2  x.  A
)  /  ( A  +  _i ) ) )
6224oveq2d 6317 . . . . 5  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  +  1 ) )  =  ( _i  x.  ( ( 2  x.  _i )  /  ( A  +  _i )
) ) )
638, 38, 8mul12i 9828 . . . . . . . 8  |-  ( _i  x.  ( 2  x.  _i ) )  =  ( 2  x.  (
_i  x.  _i )
)
64 ixi 10241 . . . . . . . . 9  |-  ( _i  x.  _i )  = 
-u 1
6564oveq2i 6312 . . . . . . . 8  |-  ( 2  x.  ( _i  x.  _i ) )  =  ( 2  x.  -u 1
)
6621negcli 9942 . . . . . . . . 9  |-  -u 1  e.  CC
6738mulm1i 10063 . . . . . . . . 9  |-  ( -u
1  x.  2 )  =  -u 2
6866, 38, 67mulcomli 9650 . . . . . . . 8  |-  ( 2  x.  -u 1 )  = 
-u 2
6963, 65, 683eqtri 2455 . . . . . . 7  |-  ( _i  x.  ( 2  x.  _i ) )  = 
-u 2
7069oveq1i 6311 . . . . . 6  |-  ( ( _i  x.  ( 2  x.  _i ) )  /  ( A  +  _i ) )  =  (
-u 2  /  ( A  +  _i )
)
7150, 5, 10, 19divassd 10418 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  ( 2  x.  _i ) )  /  ( A  +  _i ) )  =  ( _i  x.  ( ( 2  x.  _i )  /  ( A  +  _i ) ) ) )
7270, 71syl5eqr 2477 . . . . 5  |-  ( A  e.  dom arctan  ->  ( -u
2  /  ( A  +  _i ) )  =  ( _i  x.  ( ( 2  x.  _i )  /  ( A  +  _i )
) ) )
7362, 72eqtr4d 2466 . . . 4  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  +  1 ) )  =  ( -u 2  /  ( A  +  _i ) ) )
7461, 73oveq12d 6319 . . 3  |-  ( A  e.  dom arctan  ->  ( ( ( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  -  1 )  / 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  +  1 ) ) )  =  ( ( ( -u 2  x.  A )  /  ( A  +  _i )
)  /  ( -u
2  /  ( A  +  _i ) ) ) )
7538negcli 9942 . . . . . 6  |-  -u 2  e.  CC
76 mulcl 9623 . . . . . 6  |-  ( (
-u 2  e.  CC  /\  A  e.  CC )  ->  ( -u 2  x.  A )  e.  CC )
7775, 7, 76sylancr 667 . . . . 5  |-  ( A  e.  dom arctan  ->  ( -u
2  x.  A )  e.  CC )
7875a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  -u 2  e.  CC )
79 2ne0 10702 . . . . . . 7  |-  2  =/=  0
8038, 79negne0i 9949 . . . . . 6  |-  -u 2  =/=  0
8180a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  -u 2  =/=  0 )
8277, 78, 10, 81, 19divcan7d 10411 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( ( -u 2  x.  A )  /  ( A  +  _i )
)  /  ( -u
2  /  ( A  +  _i ) ) )  =  ( (
-u 2  x.  A
)  /  -u 2
) )
837, 78, 81divcan3d 10388 . . . 4  |-  ( A  e.  dom arctan  ->  ( (
-u 2  x.  A
)  /  -u 2
)  =  A )
8482, 83eqtrd 2463 . . 3  |-  ( A  e.  dom arctan  ->  ( ( ( -u 2  x.  A )  /  ( A  +  _i )
)  /  ( -u
2  /  ( A  +  _i ) ) )  =  A )
8574, 84eqtrd 2463 . 2  |-  ( A  e.  dom arctan  ->  ( ( ( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  -  1 )  / 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  +  1 ) ) )  =  A )
8630, 85eqtrd 2463 1  |-  ( A  e.  dom arctan  ->  ( tan `  (arctan `  A )
)  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1868    =/= wne 2618   dom cdm 4849   ` cfv 5597  (class class class)co 6301   CCcc 9537   0cc0 9539   1c1 9540   _ici 9541    + caddc 9542    x. cmul 9544    - cmin 9860   -ucneg 9861    / cdiv 10269   2c2 10659   expce 14101   tanctan 14105  arctancatan 23776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-of 6541  df-om 6703  df-1st 6803  df-2nd 6804  df-supp 6922  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-er 7367  df-map 7478  df-pm 7479  df-ixp 7527  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-fsupp 7886  df-fi 7927  df-sup 7958  df-inf 7959  df-oi 8027  df-card 8374  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12027  df-mod 12096  df-seq 12213  df-exp 12272  df-fac 12459  df-bc 12487  df-hash 12515  df-shft 13118  df-cj 13150  df-re 13151  df-im 13152  df-sqrt 13286  df-abs 13287  df-limsup 13513  df-clim 13539  df-rlim 13540  df-sum 13740  df-ef 14108  df-sin 14110  df-cos 14111  df-tan 14112  df-pi 14113  df-struct 15110  df-ndx 15111  df-slot 15112  df-base 15113  df-sets 15114  df-ress 15115  df-plusg 15190  df-mulr 15191  df-starv 15192  df-sca 15193  df-vsca 15194  df-ip 15195  df-tset 15196  df-ple 15197  df-ds 15199  df-unif 15200  df-hom 15201  df-cco 15202  df-rest 15308  df-topn 15309  df-0g 15327  df-gsum 15328  df-topgen 15329  df-pt 15330  df-prds 15333  df-xrs 15387  df-qtop 15393  df-imas 15394  df-xps 15397  df-mre 15479  df-mrc 15480  df-acs 15482  df-mgm 16475  df-sgrp 16514  df-mnd 16524  df-submnd 16570  df-mulg 16663  df-cntz 16958  df-cmn 17419  df-psmet 18949  df-xmet 18950  df-met 18951  df-bl 18952  df-mopn 18953  df-fbas 18954  df-fg 18955  df-cnfld 18958  df-top 19907  df-bases 19908  df-topon 19909  df-topsp 19910  df-cld 20020  df-ntr 20021  df-cls 20022  df-nei 20100  df-lp 20138  df-perf 20139  df-cn 20229  df-cnp 20230  df-haus 20317  df-tx 20563  df-hmeo 20756  df-fil 20847  df-fm 20939  df-flim 20940  df-flf 20941  df-xms 21321  df-ms 21322  df-tms 21323  df-cncf 21896  df-limc 22807  df-dv 22808  df-log 23492  df-atan 23779
This theorem is referenced by:  atantanb  23836  atanord  23839
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